We studied the critical properties of flexible polymers, modelled by self-avoiding random walks, in good solvents and homogeneous environments. By applying the PERM Monte Carlo simulation method, we generated the polymer chains on the square and the simple cubic lattice of the maximal length of N=2000 steps. We enumerated approximately the number of different polymer chain configurations of length N, and analysed its asymptotic behaviour (for large N), determined by the connectivity constant μ and the entropic critical exponent γ. Also, we studied the behaviour of the set of effective critical exponents ν_N, governing the end-to-end distance of a polymer chain of length N. We have established that in two dimensions ν_N monotonically increases with N, whereas in three dimensions it monotonically decreases when N increases. Values of ν_N, obtained for both spatial dimensions have been extrapolated in the range of very long chains. In the end, we discuss and compare our results to those obtained previously for polymers on Euclidean lattices.